Cantormängden

Cantormängden, also known as the Cantor set, is a fascinating mathematical object. It is a set of points on the line segment from 0 to 1. The construction begins with the closed interval [0, 1]. In the first step, the open middle third (1/3, 2/3) is removed, leaving two closed intervals [0, 1/3] and [2/3, 1]. In the second step, the open middle third of each of these remaining intervals is removed. Specifically, (1/9, 2/9) and (7/9, 8/9) are removed. This process is continued indefinitely.

The Cantor set is the set of points that remain after this infinite process of removing open middle thirds. Despite its construction involving the removal of intervals, the Cantor set is not empty. It contains points that are not removed at any stage. For example, 0, 1, 1/3, and 2/3 are all in the Cantor set. A key property of the Cantor set is that it is nowhere dense, meaning that every interval on the real line contains a point that is not in the Cantor set.

Another striking characteristic of the Cantor set is that it is uncountable. This means that even though it is constructed by removing parts of the interval [0, 1], it contains infinitely many points, in fact, as many points as the original interval. The Cantor set is also a fractal; it exhibits self-similarity. If you zoom in on any part of the Cantor set, you will see smaller copies of the entire set. This fractal nature makes it a classic example in the study of fractal geometry.